It turns out it is possible to normalize exactly the
resonance gain of the second-order resonator tuned by a
single coefficient [89]. This is accomplished by
placing the two zeros at
, where is the radius of
the complex-conjugate pole pair . The transfer function numerator
becomes
, yielding
the total transfer function

We see there is one more multiply-add per sample (the term )
relative to the unnormalized two-pole resonator of Eq. (10.13).
The resonance gain is now

Thus, the gain at resonance is for all resonance tunings .

Figure 10.20 shows a family of amplitude responses for the
constant resonance-gain two-pole, for various values of and
. We see an excellent improvement in the regularity of the
amplitude response as a function of tuning.

Figure:Frequency response overlays for the constant
resonance-gain two-pole filter
,
for and 10 values of
uniformly spaced from 0 to . The 5th case is
plotted using thicker lines.